Properties

Label 4730.2.a.bf.1.7
Level 4730
Weight 2
Character 4730.1
Self dual yes
Analytic conductor 37.769
Analytic rank 0
Dimension 13
CM no
Inner twists 1

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Newspace parameters

Level: \( N \) = \( 4730 = 2 \cdot 5 \cdot 11 \cdot 43 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 4730.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(37.7692401561\)
Analytic rank: \(0\)
Dimension: \(13\)
Coefficient field: \(\mathbb{Q}[x]/(x^{13} - \cdots)\)
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.7
Root \(0.0802958\)
Character \(\chi\) = 4730.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000 q^{2} -0.0802958 q^{3} +1.00000 q^{4} -1.00000 q^{5} -0.0802958 q^{6} +2.53237 q^{7} +1.00000 q^{8} -2.99355 q^{9} +O(q^{10})\) \(q+1.00000 q^{2} -0.0802958 q^{3} +1.00000 q^{4} -1.00000 q^{5} -0.0802958 q^{6} +2.53237 q^{7} +1.00000 q^{8} -2.99355 q^{9} -1.00000 q^{10} +1.00000 q^{11} -0.0802958 q^{12} +5.53942 q^{13} +2.53237 q^{14} +0.0802958 q^{15} +1.00000 q^{16} -1.79237 q^{17} -2.99355 q^{18} -1.35154 q^{19} -1.00000 q^{20} -0.203338 q^{21} +1.00000 q^{22} +2.94296 q^{23} -0.0802958 q^{24} +1.00000 q^{25} +5.53942 q^{26} +0.481257 q^{27} +2.53237 q^{28} +5.51381 q^{29} +0.0802958 q^{30} -4.14071 q^{31} +1.00000 q^{32} -0.0802958 q^{33} -1.79237 q^{34} -2.53237 q^{35} -2.99355 q^{36} +11.9002 q^{37} -1.35154 q^{38} -0.444792 q^{39} -1.00000 q^{40} -8.42383 q^{41} -0.203338 q^{42} +1.00000 q^{43} +1.00000 q^{44} +2.99355 q^{45} +2.94296 q^{46} -5.26799 q^{47} -0.0802958 q^{48} -0.587126 q^{49} +1.00000 q^{50} +0.143919 q^{51} +5.53942 q^{52} +3.73022 q^{53} +0.481257 q^{54} -1.00000 q^{55} +2.53237 q^{56} +0.108523 q^{57} +5.51381 q^{58} -6.86311 q^{59} +0.0802958 q^{60} +3.71484 q^{61} -4.14071 q^{62} -7.58077 q^{63} +1.00000 q^{64} -5.53942 q^{65} -0.0802958 q^{66} +11.0507 q^{67} -1.79237 q^{68} -0.236307 q^{69} -2.53237 q^{70} -9.33354 q^{71} -2.99355 q^{72} +14.5726 q^{73} +11.9002 q^{74} -0.0802958 q^{75} -1.35154 q^{76} +2.53237 q^{77} -0.444792 q^{78} +13.5992 q^{79} -1.00000 q^{80} +8.94201 q^{81} -8.42383 q^{82} +2.07886 q^{83} -0.203338 q^{84} +1.79237 q^{85} +1.00000 q^{86} -0.442736 q^{87} +1.00000 q^{88} -4.41684 q^{89} +2.99355 q^{90} +14.0278 q^{91} +2.94296 q^{92} +0.332482 q^{93} -5.26799 q^{94} +1.35154 q^{95} -0.0802958 q^{96} -11.3450 q^{97} -0.587126 q^{98} -2.99355 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 13q + 13q^{2} + 13q^{4} - 13q^{5} + 7q^{7} + 13q^{8} + 25q^{9} + O(q^{10}) \) \( 13q + 13q^{2} + 13q^{4} - 13q^{5} + 7q^{7} + 13q^{8} + 25q^{9} - 13q^{10} + 13q^{11} + 11q^{13} + 7q^{14} + 13q^{16} - 2q^{17} + 25q^{18} + 7q^{19} - 13q^{20} + 12q^{21} + 13q^{22} + 12q^{23} + 13q^{25} + 11q^{26} - 15q^{27} + 7q^{28} + 14q^{29} + 20q^{31} + 13q^{32} - 2q^{34} - 7q^{35} + 25q^{36} + 17q^{37} + 7q^{38} - 4q^{39} - 13q^{40} + 9q^{41} + 12q^{42} + 13q^{43} + 13q^{44} - 25q^{45} + 12q^{46} - 9q^{47} + 30q^{49} + 13q^{50} - 3q^{51} + 11q^{52} + 22q^{53} - 15q^{54} - 13q^{55} + 7q^{56} + 17q^{57} + 14q^{58} + 19q^{59} + 2q^{61} + 20q^{62} + 12q^{63} + 13q^{64} - 11q^{65} + 9q^{67} - 2q^{68} - 6q^{69} - 7q^{70} + 6q^{71} + 25q^{72} + 7q^{73} + 17q^{74} + 7q^{76} + 7q^{77} - 4q^{78} + 50q^{79} - 13q^{80} + 85q^{81} + 9q^{82} + q^{83} + 12q^{84} + 2q^{85} + 13q^{86} - 21q^{87} + 13q^{88} + 5q^{89} - 25q^{90} + 5q^{91} + 12q^{92} + 3q^{93} - 9q^{94} - 7q^{95} + 20q^{97} + 30q^{98} + 25q^{99} + O(q^{100}) \)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).

Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.00000 0.707107
\(3\) −0.0802958 −0.0463588 −0.0231794 0.999731i \(-0.507379\pi\)
−0.0231794 + 0.999731i \(0.507379\pi\)
\(4\) 1.00000 0.500000
\(5\) −1.00000 −0.447214
\(6\) −0.0802958 −0.0327806
\(7\) 2.53237 0.957144 0.478572 0.878048i \(-0.341155\pi\)
0.478572 + 0.878048i \(0.341155\pi\)
\(8\) 1.00000 0.353553
\(9\) −2.99355 −0.997851
\(10\) −1.00000 −0.316228
\(11\) 1.00000 0.301511
\(12\) −0.0802958 −0.0231794
\(13\) 5.53942 1.53636 0.768179 0.640235i \(-0.221163\pi\)
0.768179 + 0.640235i \(0.221163\pi\)
\(14\) 2.53237 0.676803
\(15\) 0.0802958 0.0207323
\(16\) 1.00000 0.250000
\(17\) −1.79237 −0.434713 −0.217356 0.976092i \(-0.569743\pi\)
−0.217356 + 0.976092i \(0.569743\pi\)
\(18\) −2.99355 −0.705587
\(19\) −1.35154 −0.310065 −0.155032 0.987909i \(-0.549548\pi\)
−0.155032 + 0.987909i \(0.549548\pi\)
\(20\) −1.00000 −0.223607
\(21\) −0.203338 −0.0443720
\(22\) 1.00000 0.213201
\(23\) 2.94296 0.613650 0.306825 0.951766i \(-0.400733\pi\)
0.306825 + 0.951766i \(0.400733\pi\)
\(24\) −0.0802958 −0.0163903
\(25\) 1.00000 0.200000
\(26\) 5.53942 1.08637
\(27\) 0.481257 0.0926180
\(28\) 2.53237 0.478572
\(29\) 5.51381 1.02389 0.511944 0.859019i \(-0.328925\pi\)
0.511944 + 0.859019i \(0.328925\pi\)
\(30\) 0.0802958 0.0146599
\(31\) −4.14071 −0.743693 −0.371847 0.928294i \(-0.621275\pi\)
−0.371847 + 0.928294i \(0.621275\pi\)
\(32\) 1.00000 0.176777
\(33\) −0.0802958 −0.0139777
\(34\) −1.79237 −0.307388
\(35\) −2.53237 −0.428048
\(36\) −2.99355 −0.498925
\(37\) 11.9002 1.95638 0.978188 0.207719i \(-0.0666040\pi\)
0.978188 + 0.207719i \(0.0666040\pi\)
\(38\) −1.35154 −0.219249
\(39\) −0.444792 −0.0712237
\(40\) −1.00000 −0.158114
\(41\) −8.42383 −1.31558 −0.657791 0.753201i \(-0.728509\pi\)
−0.657791 + 0.753201i \(0.728509\pi\)
\(42\) −0.203338 −0.0313758
\(43\) 1.00000 0.152499
\(44\) 1.00000 0.150756
\(45\) 2.99355 0.446252
\(46\) 2.94296 0.433916
\(47\) −5.26799 −0.768416 −0.384208 0.923247i \(-0.625525\pi\)
−0.384208 + 0.923247i \(0.625525\pi\)
\(48\) −0.0802958 −0.0115897
\(49\) −0.587126 −0.0838752
\(50\) 1.00000 0.141421
\(51\) 0.143919 0.0201528
\(52\) 5.53942 0.768179
\(53\) 3.73022 0.512385 0.256192 0.966626i \(-0.417532\pi\)
0.256192 + 0.966626i \(0.417532\pi\)
\(54\) 0.481257 0.0654908
\(55\) −1.00000 −0.134840
\(56\) 2.53237 0.338402
\(57\) 0.108523 0.0143742
\(58\) 5.51381 0.723998
\(59\) −6.86311 −0.893500 −0.446750 0.894659i \(-0.647419\pi\)
−0.446750 + 0.894659i \(0.647419\pi\)
\(60\) 0.0802958 0.0103661
\(61\) 3.71484 0.475637 0.237818 0.971310i \(-0.423568\pi\)
0.237818 + 0.971310i \(0.423568\pi\)
\(62\) −4.14071 −0.525871
\(63\) −7.58077 −0.955087
\(64\) 1.00000 0.125000
\(65\) −5.53942 −0.687080
\(66\) −0.0802958 −0.00988373
\(67\) 11.0507 1.35006 0.675031 0.737789i \(-0.264130\pi\)
0.675031 + 0.737789i \(0.264130\pi\)
\(68\) −1.79237 −0.217356
\(69\) −0.236307 −0.0284481
\(70\) −2.53237 −0.302676
\(71\) −9.33354 −1.10769 −0.553844 0.832621i \(-0.686839\pi\)
−0.553844 + 0.832621i \(0.686839\pi\)
\(72\) −2.99355 −0.352794
\(73\) 14.5726 1.70559 0.852795 0.522247i \(-0.174906\pi\)
0.852795 + 0.522247i \(0.174906\pi\)
\(74\) 11.9002 1.38337
\(75\) −0.0802958 −0.00927176
\(76\) −1.35154 −0.155032
\(77\) 2.53237 0.288590
\(78\) −0.444792 −0.0503628
\(79\) 13.5992 1.53003 0.765014 0.644014i \(-0.222732\pi\)
0.765014 + 0.644014i \(0.222732\pi\)
\(80\) −1.00000 −0.111803
\(81\) 8.94201 0.993557
\(82\) −8.42383 −0.930256
\(83\) 2.07886 0.228184 0.114092 0.993470i \(-0.463604\pi\)
0.114092 + 0.993470i \(0.463604\pi\)
\(84\) −0.203338 −0.0221860
\(85\) 1.79237 0.194409
\(86\) 1.00000 0.107833
\(87\) −0.442736 −0.0474662
\(88\) 1.00000 0.106600
\(89\) −4.41684 −0.468184 −0.234092 0.972214i \(-0.575212\pi\)
−0.234092 + 0.972214i \(0.575212\pi\)
\(90\) 2.99355 0.315548
\(91\) 14.0278 1.47052
\(92\) 2.94296 0.306825
\(93\) 0.332482 0.0344767
\(94\) −5.26799 −0.543352
\(95\) 1.35154 0.138665
\(96\) −0.0802958 −0.00819515
\(97\) −11.3450 −1.15191 −0.575955 0.817481i \(-0.695370\pi\)
−0.575955 + 0.817481i \(0.695370\pi\)
\(98\) −0.587126 −0.0593087
\(99\) −2.99355 −0.300863
\(100\) 1.00000 0.100000
\(101\) −10.8959 −1.08418 −0.542089 0.840321i \(-0.682367\pi\)
−0.542089 + 0.840321i \(0.682367\pi\)
\(102\) 0.143919 0.0142502
\(103\) 6.15015 0.605992 0.302996 0.952992i \(-0.402013\pi\)
0.302996 + 0.952992i \(0.402013\pi\)
\(104\) 5.53942 0.543185
\(105\) 0.203338 0.0198438
\(106\) 3.73022 0.362311
\(107\) 15.8106 1.52847 0.764233 0.644941i \(-0.223118\pi\)
0.764233 + 0.644941i \(0.223118\pi\)
\(108\) 0.481257 0.0463090
\(109\) 7.13096 0.683022 0.341511 0.939878i \(-0.389061\pi\)
0.341511 + 0.939878i \(0.389061\pi\)
\(110\) −1.00000 −0.0953463
\(111\) −0.955534 −0.0906953
\(112\) 2.53237 0.239286
\(113\) 18.8176 1.77021 0.885105 0.465392i \(-0.154086\pi\)
0.885105 + 0.465392i \(0.154086\pi\)
\(114\) 0.108523 0.0101641
\(115\) −2.94296 −0.274433
\(116\) 5.51381 0.511944
\(117\) −16.5825 −1.53306
\(118\) −6.86311 −0.631800
\(119\) −4.53893 −0.416083
\(120\) 0.0802958 0.00732997
\(121\) 1.00000 0.0909091
\(122\) 3.71484 0.336326
\(123\) 0.676398 0.0609888
\(124\) −4.14071 −0.371847
\(125\) −1.00000 −0.0894427
\(126\) −7.58077 −0.675349
\(127\) −1.63100 −0.144728 −0.0723640 0.997378i \(-0.523054\pi\)
−0.0723640 + 0.997378i \(0.523054\pi\)
\(128\) 1.00000 0.0883883
\(129\) −0.0802958 −0.00706965
\(130\) −5.53942 −0.485839
\(131\) 15.3193 1.33846 0.669228 0.743057i \(-0.266625\pi\)
0.669228 + 0.743057i \(0.266625\pi\)
\(132\) −0.0802958 −0.00698885
\(133\) −3.42259 −0.296777
\(134\) 11.0507 0.954638
\(135\) −0.481257 −0.0414200
\(136\) −1.79237 −0.153694
\(137\) 1.72740 0.147581 0.0737907 0.997274i \(-0.476490\pi\)
0.0737907 + 0.997274i \(0.476490\pi\)
\(138\) −0.236307 −0.0201158
\(139\) −0.527955 −0.0447806 −0.0223903 0.999749i \(-0.507128\pi\)
−0.0223903 + 0.999749i \(0.507128\pi\)
\(140\) −2.53237 −0.214024
\(141\) 0.422998 0.0356228
\(142\) −9.33354 −0.783253
\(143\) 5.53942 0.463229
\(144\) −2.99355 −0.249463
\(145\) −5.51381 −0.457897
\(146\) 14.5726 1.20603
\(147\) 0.0471438 0.00388835
\(148\) 11.9002 0.978188
\(149\) 0.606138 0.0496568 0.0248284 0.999692i \(-0.492096\pi\)
0.0248284 + 0.999692i \(0.492096\pi\)
\(150\) −0.0802958 −0.00655612
\(151\) −14.0485 −1.14325 −0.571625 0.820515i \(-0.693687\pi\)
−0.571625 + 0.820515i \(0.693687\pi\)
\(152\) −1.35154 −0.109624
\(153\) 5.36554 0.433779
\(154\) 2.53237 0.204064
\(155\) 4.14071 0.332590
\(156\) −0.444792 −0.0356119
\(157\) −12.7865 −1.02047 −0.510235 0.860035i \(-0.670442\pi\)
−0.510235 + 0.860035i \(0.670442\pi\)
\(158\) 13.5992 1.08189
\(159\) −0.299521 −0.0237535
\(160\) −1.00000 −0.0790569
\(161\) 7.45265 0.587351
\(162\) 8.94201 0.702551
\(163\) 1.14615 0.0897733 0.0448866 0.998992i \(-0.485707\pi\)
0.0448866 + 0.998992i \(0.485707\pi\)
\(164\) −8.42383 −0.657791
\(165\) 0.0802958 0.00625102
\(166\) 2.07886 0.161351
\(167\) 18.8684 1.46008 0.730041 0.683403i \(-0.239501\pi\)
0.730041 + 0.683403i \(0.239501\pi\)
\(168\) −0.203338 −0.0156879
\(169\) 17.6852 1.36040
\(170\) 1.79237 0.137468
\(171\) 4.04591 0.309398
\(172\) 1.00000 0.0762493
\(173\) −21.7578 −1.65421 −0.827107 0.562044i \(-0.810015\pi\)
−0.827107 + 0.562044i \(0.810015\pi\)
\(174\) −0.442736 −0.0335637
\(175\) 2.53237 0.191429
\(176\) 1.00000 0.0753778
\(177\) 0.551078 0.0414216
\(178\) −4.41684 −0.331056
\(179\) 10.6020 0.792428 0.396214 0.918158i \(-0.370324\pi\)
0.396214 + 0.918158i \(0.370324\pi\)
\(180\) 2.99355 0.223126
\(181\) −21.1772 −1.57409 −0.787043 0.616898i \(-0.788389\pi\)
−0.787043 + 0.616898i \(0.788389\pi\)
\(182\) 14.0278 1.03981
\(183\) −0.298286 −0.0220499
\(184\) 2.94296 0.216958
\(185\) −11.9002 −0.874918
\(186\) 0.332482 0.0243787
\(187\) −1.79237 −0.131071
\(188\) −5.26799 −0.384208
\(189\) 1.21872 0.0886487
\(190\) 1.35154 0.0980511
\(191\) −17.3295 −1.25392 −0.626958 0.779053i \(-0.715700\pi\)
−0.626958 + 0.779053i \(0.715700\pi\)
\(192\) −0.0802958 −0.00579485
\(193\) −18.5364 −1.33428 −0.667140 0.744932i \(-0.732482\pi\)
−0.667140 + 0.744932i \(0.732482\pi\)
\(194\) −11.3450 −0.814523
\(195\) 0.444792 0.0318522
\(196\) −0.587126 −0.0419376
\(197\) 16.1741 1.15236 0.576178 0.817324i \(-0.304543\pi\)
0.576178 + 0.817324i \(0.304543\pi\)
\(198\) −2.99355 −0.212743
\(199\) 19.3164 1.36931 0.684653 0.728869i \(-0.259954\pi\)
0.684653 + 0.728869i \(0.259954\pi\)
\(200\) 1.00000 0.0707107
\(201\) −0.887327 −0.0625872
\(202\) −10.8959 −0.766630
\(203\) 13.9630 0.980009
\(204\) 0.143919 0.0100764
\(205\) 8.42383 0.588346
\(206\) 6.15015 0.428501
\(207\) −8.80991 −0.612331
\(208\) 5.53942 0.384090
\(209\) −1.35154 −0.0934880
\(210\) 0.203338 0.0140317
\(211\) 11.3192 0.779247 0.389624 0.920974i \(-0.372605\pi\)
0.389624 + 0.920974i \(0.372605\pi\)
\(212\) 3.73022 0.256192
\(213\) 0.749444 0.0513511
\(214\) 15.8106 1.08079
\(215\) −1.00000 −0.0681994
\(216\) 0.481257 0.0327454
\(217\) −10.4858 −0.711822
\(218\) 7.13096 0.482970
\(219\) −1.17012 −0.0790691
\(220\) −1.00000 −0.0674200
\(221\) −9.92867 −0.667875
\(222\) −0.955534 −0.0641312
\(223\) 8.13650 0.544860 0.272430 0.962176i \(-0.412173\pi\)
0.272430 + 0.962176i \(0.412173\pi\)
\(224\) 2.53237 0.169201
\(225\) −2.99355 −0.199570
\(226\) 18.8176 1.25173
\(227\) 26.4976 1.75871 0.879353 0.476170i \(-0.157975\pi\)
0.879353 + 0.476170i \(0.157975\pi\)
\(228\) 0.108523 0.00718711
\(229\) 20.7862 1.37359 0.686794 0.726852i \(-0.259017\pi\)
0.686794 + 0.726852i \(0.259017\pi\)
\(230\) −2.94296 −0.194053
\(231\) −0.203338 −0.0133787
\(232\) 5.51381 0.361999
\(233\) 24.6597 1.61551 0.807754 0.589519i \(-0.200683\pi\)
0.807754 + 0.589519i \(0.200683\pi\)
\(234\) −16.5825 −1.08403
\(235\) 5.26799 0.343646
\(236\) −6.86311 −0.446750
\(237\) −1.09196 −0.0709303
\(238\) −4.53893 −0.294215
\(239\) −25.4210 −1.64435 −0.822173 0.569238i \(-0.807238\pi\)
−0.822173 + 0.569238i \(0.807238\pi\)
\(240\) 0.0802958 0.00518307
\(241\) 0.339825 0.0218901 0.0109450 0.999940i \(-0.496516\pi\)
0.0109450 + 0.999940i \(0.496516\pi\)
\(242\) 1.00000 0.0642824
\(243\) −2.16178 −0.138678
\(244\) 3.71484 0.237818
\(245\) 0.587126 0.0375101
\(246\) 0.676398 0.0431256
\(247\) −7.48675 −0.476370
\(248\) −4.14071 −0.262935
\(249\) −0.166923 −0.0105783
\(250\) −1.00000 −0.0632456
\(251\) −14.0703 −0.888112 −0.444056 0.895999i \(-0.646461\pi\)
−0.444056 + 0.895999i \(0.646461\pi\)
\(252\) −7.58077 −0.477544
\(253\) 2.94296 0.185022
\(254\) −1.63100 −0.102338
\(255\) −0.143919 −0.00901259
\(256\) 1.00000 0.0625000
\(257\) 3.24329 0.202311 0.101156 0.994871i \(-0.467746\pi\)
0.101156 + 0.994871i \(0.467746\pi\)
\(258\) −0.0802958 −0.00499900
\(259\) 30.1356 1.87253
\(260\) −5.53942 −0.343540
\(261\) −16.5059 −1.02169
\(262\) 15.3193 0.946432
\(263\) −20.8739 −1.28714 −0.643569 0.765388i \(-0.722547\pi\)
−0.643569 + 0.765388i \(0.722547\pi\)
\(264\) −0.0802958 −0.00494186
\(265\) −3.73022 −0.229145
\(266\) −3.42259 −0.209853
\(267\) 0.354654 0.0217045
\(268\) 11.0507 0.675031
\(269\) 7.57253 0.461706 0.230853 0.972989i \(-0.425848\pi\)
0.230853 + 0.972989i \(0.425848\pi\)
\(270\) −0.481257 −0.0292884
\(271\) −12.0129 −0.729731 −0.364865 0.931060i \(-0.618885\pi\)
−0.364865 + 0.931060i \(0.618885\pi\)
\(272\) −1.79237 −0.108678
\(273\) −1.12638 −0.0681714
\(274\) 1.72740 0.104356
\(275\) 1.00000 0.0603023
\(276\) −0.236307 −0.0142240
\(277\) 26.4077 1.58669 0.793343 0.608775i \(-0.208339\pi\)
0.793343 + 0.608775i \(0.208339\pi\)
\(278\) −0.527955 −0.0316647
\(279\) 12.3954 0.742095
\(280\) −2.53237 −0.151338
\(281\) 4.35199 0.259618 0.129809 0.991539i \(-0.458564\pi\)
0.129809 + 0.991539i \(0.458564\pi\)
\(282\) 0.422998 0.0251891
\(283\) 30.9015 1.83691 0.918453 0.395530i \(-0.129439\pi\)
0.918453 + 0.395530i \(0.129439\pi\)
\(284\) −9.33354 −0.553844
\(285\) −0.108523 −0.00642835
\(286\) 5.53942 0.327553
\(287\) −21.3322 −1.25920
\(288\) −2.99355 −0.176397
\(289\) −13.7874 −0.811025
\(290\) −5.51381 −0.323782
\(291\) 0.910955 0.0534012
\(292\) 14.5726 0.852795
\(293\) 19.5789 1.14381 0.571905 0.820320i \(-0.306205\pi\)
0.571905 + 0.820320i \(0.306205\pi\)
\(294\) 0.0471438 0.00274948
\(295\) 6.86311 0.399585
\(296\) 11.9002 0.691684
\(297\) 0.481257 0.0279254
\(298\) 0.606138 0.0351126
\(299\) 16.3023 0.942786
\(300\) −0.0802958 −0.00463588
\(301\) 2.53237 0.145963
\(302\) −14.0485 −0.808399
\(303\) 0.874892 0.0502612
\(304\) −1.35154 −0.0775162
\(305\) −3.71484 −0.212711
\(306\) 5.36554 0.306728
\(307\) 23.5744 1.34546 0.672730 0.739888i \(-0.265122\pi\)
0.672730 + 0.739888i \(0.265122\pi\)
\(308\) 2.53237 0.144295
\(309\) −0.493831 −0.0280931
\(310\) 4.14071 0.235176
\(311\) −27.0824 −1.53570 −0.767852 0.640628i \(-0.778674\pi\)
−0.767852 + 0.640628i \(0.778674\pi\)
\(312\) −0.444792 −0.0251814
\(313\) −15.2582 −0.862447 −0.431223 0.902245i \(-0.641918\pi\)
−0.431223 + 0.902245i \(0.641918\pi\)
\(314\) −12.7865 −0.721581
\(315\) 7.58077 0.427128
\(316\) 13.5992 0.765014
\(317\) 4.50710 0.253144 0.126572 0.991957i \(-0.459603\pi\)
0.126572 + 0.991957i \(0.459603\pi\)
\(318\) −0.299521 −0.0167963
\(319\) 5.51381 0.308714
\(320\) −1.00000 −0.0559017
\(321\) −1.26952 −0.0708578
\(322\) 7.45265 0.415320
\(323\) 2.42246 0.134789
\(324\) 8.94201 0.496779
\(325\) 5.53942 0.307272
\(326\) 1.14615 0.0634793
\(327\) −0.572586 −0.0316641
\(328\) −8.42383 −0.465128
\(329\) −13.3405 −0.735484
\(330\) 0.0802958 0.00442014
\(331\) −15.3588 −0.844198 −0.422099 0.906550i \(-0.638707\pi\)
−0.422099 + 0.906550i \(0.638707\pi\)
\(332\) 2.07886 0.114092
\(333\) −35.6238 −1.95217
\(334\) 18.8684 1.03243
\(335\) −11.0507 −0.603766
\(336\) −0.203338 −0.0110930
\(337\) −20.9110 −1.13910 −0.569549 0.821958i \(-0.692882\pi\)
−0.569549 + 0.821958i \(0.692882\pi\)
\(338\) 17.6852 0.961946
\(339\) −1.51097 −0.0820648
\(340\) 1.79237 0.0972047
\(341\) −4.14071 −0.224232
\(342\) 4.04591 0.218778
\(343\) −19.2134 −1.03742
\(344\) 1.00000 0.0539164
\(345\) 0.236307 0.0127224
\(346\) −21.7578 −1.16971
\(347\) −15.2696 −0.819715 −0.409858 0.912150i \(-0.634422\pi\)
−0.409858 + 0.912150i \(0.634422\pi\)
\(348\) −0.442736 −0.0237331
\(349\) −7.74515 −0.414588 −0.207294 0.978279i \(-0.566466\pi\)
−0.207294 + 0.978279i \(0.566466\pi\)
\(350\) 2.53237 0.135361
\(351\) 2.66588 0.142294
\(352\) 1.00000 0.0533002
\(353\) −27.2898 −1.45249 −0.726246 0.687435i \(-0.758737\pi\)
−0.726246 + 0.687435i \(0.758737\pi\)
\(354\) 0.551078 0.0292895
\(355\) 9.33354 0.495373
\(356\) −4.41684 −0.234092
\(357\) 0.364457 0.0192891
\(358\) 10.6020 0.560331
\(359\) −22.5389 −1.18956 −0.594778 0.803890i \(-0.702760\pi\)
−0.594778 + 0.803890i \(0.702760\pi\)
\(360\) 2.99355 0.157774
\(361\) −17.1733 −0.903860
\(362\) −21.1772 −1.11305
\(363\) −0.0802958 −0.00421444
\(364\) 14.0278 0.735258
\(365\) −14.5726 −0.762763
\(366\) −0.298286 −0.0155917
\(367\) −3.64335 −0.190181 −0.0950907 0.995469i \(-0.530314\pi\)
−0.0950907 + 0.995469i \(0.530314\pi\)
\(368\) 2.94296 0.153412
\(369\) 25.2172 1.31275
\(370\) −11.9002 −0.618661
\(371\) 9.44627 0.490426
\(372\) 0.332482 0.0172384
\(373\) −22.3092 −1.15513 −0.577563 0.816346i \(-0.695996\pi\)
−0.577563 + 0.816346i \(0.695996\pi\)
\(374\) −1.79237 −0.0926811
\(375\) 0.0802958 0.00414646
\(376\) −5.26799 −0.271676
\(377\) 30.5433 1.57306
\(378\) 1.21872 0.0626841
\(379\) −4.46027 −0.229108 −0.114554 0.993417i \(-0.536544\pi\)
−0.114554 + 0.993417i \(0.536544\pi\)
\(380\) 1.35154 0.0693326
\(381\) 0.130963 0.00670942
\(382\) −17.3295 −0.886652
\(383\) −20.1980 −1.03207 −0.516034 0.856568i \(-0.672592\pi\)
−0.516034 + 0.856568i \(0.672592\pi\)
\(384\) −0.0802958 −0.00409758
\(385\) −2.53237 −0.129061
\(386\) −18.5364 −0.943479
\(387\) −2.99355 −0.152171
\(388\) −11.3450 −0.575955
\(389\) 19.0027 0.963474 0.481737 0.876316i \(-0.340006\pi\)
0.481737 + 0.876316i \(0.340006\pi\)
\(390\) 0.444792 0.0225229
\(391\) −5.27487 −0.266761
\(392\) −0.587126 −0.0296543
\(393\) −1.23008 −0.0620492
\(394\) 16.1741 0.814839
\(395\) −13.5992 −0.684249
\(396\) −2.99355 −0.150432
\(397\) −1.53517 −0.0770480 −0.0385240 0.999258i \(-0.512266\pi\)
−0.0385240 + 0.999258i \(0.512266\pi\)
\(398\) 19.3164 0.968245
\(399\) 0.274820 0.0137582
\(400\) 1.00000 0.0500000
\(401\) 9.82030 0.490403 0.245201 0.969472i \(-0.421146\pi\)
0.245201 + 0.969472i \(0.421146\pi\)
\(402\) −0.887327 −0.0442559
\(403\) −22.9371 −1.14258
\(404\) −10.8959 −0.542089
\(405\) −8.94201 −0.444332
\(406\) 13.9630 0.692971
\(407\) 11.9002 0.589870
\(408\) 0.143919 0.00712508
\(409\) 20.0718 0.992485 0.496242 0.868184i \(-0.334713\pi\)
0.496242 + 0.868184i \(0.334713\pi\)
\(410\) 8.42383 0.416023
\(411\) −0.138703 −0.00684170
\(412\) 6.15015 0.302996
\(413\) −17.3799 −0.855208
\(414\) −8.80991 −0.432984
\(415\) −2.07886 −0.102047
\(416\) 5.53942 0.271592
\(417\) 0.0423926 0.00207597
\(418\) −1.35154 −0.0661060
\(419\) −34.7946 −1.69983 −0.849915 0.526920i \(-0.823347\pi\)
−0.849915 + 0.526920i \(0.823347\pi\)
\(420\) 0.203338 0.00992189
\(421\) −10.3465 −0.504259 −0.252130 0.967693i \(-0.581131\pi\)
−0.252130 + 0.967693i \(0.581131\pi\)
\(422\) 11.3192 0.551011
\(423\) 15.7700 0.766764
\(424\) 3.73022 0.181155
\(425\) −1.79237 −0.0869426
\(426\) 0.749444 0.0363107
\(427\) 9.40734 0.455253
\(428\) 15.8106 0.764233
\(429\) −0.444792 −0.0214748
\(430\) −1.00000 −0.0482243
\(431\) −10.2756 −0.494960 −0.247480 0.968893i \(-0.579602\pi\)
−0.247480 + 0.968893i \(0.579602\pi\)
\(432\) 0.481257 0.0231545
\(433\) 24.6166 1.18300 0.591499 0.806305i \(-0.298536\pi\)
0.591499 + 0.806305i \(0.298536\pi\)
\(434\) −10.4858 −0.503334
\(435\) 0.442736 0.0212275
\(436\) 7.13096 0.341511
\(437\) −3.97753 −0.190271
\(438\) −1.17012 −0.0559103
\(439\) 2.26151 0.107936 0.0539680 0.998543i \(-0.482813\pi\)
0.0539680 + 0.998543i \(0.482813\pi\)
\(440\) −1.00000 −0.0476731
\(441\) 1.75759 0.0836949
\(442\) −9.92867 −0.472259
\(443\) −28.6143 −1.35950 −0.679752 0.733442i \(-0.737913\pi\)
−0.679752 + 0.733442i \(0.737913\pi\)
\(444\) −0.955534 −0.0453476
\(445\) 4.41684 0.209378
\(446\) 8.13650 0.385274
\(447\) −0.0486703 −0.00230203
\(448\) 2.53237 0.119643
\(449\) 10.5148 0.496222 0.248111 0.968732i \(-0.420190\pi\)
0.248111 + 0.968732i \(0.420190\pi\)
\(450\) −2.99355 −0.141117
\(451\) −8.42383 −0.396663
\(452\) 18.8176 0.885105
\(453\) 1.12803 0.0529997
\(454\) 26.4976 1.24359
\(455\) −14.0278 −0.657635
\(456\) 0.108523 0.00508206
\(457\) 34.8469 1.63007 0.815035 0.579412i \(-0.196718\pi\)
0.815035 + 0.579412i \(0.196718\pi\)
\(458\) 20.7862 0.971273
\(459\) −0.862589 −0.0402622
\(460\) −2.94296 −0.137216
\(461\) −20.8632 −0.971695 −0.485847 0.874044i \(-0.661489\pi\)
−0.485847 + 0.874044i \(0.661489\pi\)
\(462\) −0.203338 −0.00946015
\(463\) 7.25956 0.337380 0.168690 0.985669i \(-0.446046\pi\)
0.168690 + 0.985669i \(0.446046\pi\)
\(464\) 5.51381 0.255972
\(465\) −0.332482 −0.0154185
\(466\) 24.6597 1.14234
\(467\) −14.0704 −0.651100 −0.325550 0.945525i \(-0.605549\pi\)
−0.325550 + 0.945525i \(0.605549\pi\)
\(468\) −16.5825 −0.766528
\(469\) 27.9845 1.29220
\(470\) 5.26799 0.242994
\(471\) 1.02670 0.0473078
\(472\) −6.86311 −0.315900
\(473\) 1.00000 0.0459800
\(474\) −1.09196 −0.0501553
\(475\) −1.35154 −0.0620129
\(476\) −4.53893 −0.208041
\(477\) −11.1666 −0.511284
\(478\) −25.4210 −1.16273
\(479\) 37.2717 1.70299 0.851495 0.524363i \(-0.175697\pi\)
0.851495 + 0.524363i \(0.175697\pi\)
\(480\) 0.0802958 0.00366498
\(481\) 65.9201 3.00570
\(482\) 0.339825 0.0154786
\(483\) −0.598417 −0.0272289
\(484\) 1.00000 0.0454545
\(485\) 11.3450 0.515150
\(486\) −2.16178 −0.0980602
\(487\) −27.3833 −1.24086 −0.620428 0.784263i \(-0.713041\pi\)
−0.620428 + 0.784263i \(0.713041\pi\)
\(488\) 3.71484 0.168163
\(489\) −0.0920309 −0.00416178
\(490\) 0.587126 0.0265237
\(491\) 18.8903 0.852508 0.426254 0.904603i \(-0.359833\pi\)
0.426254 + 0.904603i \(0.359833\pi\)
\(492\) 0.676398 0.0304944
\(493\) −9.88277 −0.445097
\(494\) −7.48675 −0.336845
\(495\) 2.99355 0.134550
\(496\) −4.14071 −0.185923
\(497\) −23.6359 −1.06022
\(498\) −0.166923 −0.00748002
\(499\) 7.35085 0.329069 0.164535 0.986371i \(-0.447388\pi\)
0.164535 + 0.986371i \(0.447388\pi\)
\(500\) −1.00000 −0.0447214
\(501\) −1.51505 −0.0676876
\(502\) −14.0703 −0.627990
\(503\) 23.0497 1.02773 0.513867 0.857870i \(-0.328212\pi\)
0.513867 + 0.857870i \(0.328212\pi\)
\(504\) −7.58077 −0.337674
\(505\) 10.8959 0.484859
\(506\) 2.94296 0.130831
\(507\) −1.42004 −0.0630664
\(508\) −1.63100 −0.0723640
\(509\) −5.43536 −0.240918 −0.120459 0.992718i \(-0.538437\pi\)
−0.120459 + 0.992718i \(0.538437\pi\)
\(510\) −0.143919 −0.00637286
\(511\) 36.9030 1.63249
\(512\) 1.00000 0.0441942
\(513\) −0.650438 −0.0287176
\(514\) 3.24329 0.143056
\(515\) −6.15015 −0.271008
\(516\) −0.0802958 −0.00353483
\(517\) −5.26799 −0.231686
\(518\) 30.1356 1.32408
\(519\) 1.74706 0.0766874
\(520\) −5.53942 −0.242920
\(521\) −14.0472 −0.615418 −0.307709 0.951481i \(-0.599562\pi\)
−0.307709 + 0.951481i \(0.599562\pi\)
\(522\) −16.5059 −0.722442
\(523\) −12.4419 −0.544047 −0.272024 0.962291i \(-0.587693\pi\)
−0.272024 + 0.962291i \(0.587693\pi\)
\(524\) 15.3193 0.669228
\(525\) −0.203338 −0.00887441
\(526\) −20.8739 −0.910145
\(527\) 7.42167 0.323293
\(528\) −0.0802958 −0.00349443
\(529\) −14.3390 −0.623434
\(530\) −3.73022 −0.162030
\(531\) 20.5451 0.891580
\(532\) −3.42259 −0.148388
\(533\) −46.6631 −2.02120
\(534\) 0.354654 0.0153474
\(535\) −15.8106 −0.683551
\(536\) 11.0507 0.477319
\(537\) −0.851293 −0.0367360
\(538\) 7.57253 0.326475
\(539\) −0.587126 −0.0252893
\(540\) −0.481257 −0.0207100
\(541\) −22.6114 −0.972141 −0.486071 0.873919i \(-0.661570\pi\)
−0.486071 + 0.873919i \(0.661570\pi\)
\(542\) −12.0129 −0.515998
\(543\) 1.70044 0.0729727
\(544\) −1.79237 −0.0768471
\(545\) −7.13096 −0.305457
\(546\) −1.12638 −0.0482044
\(547\) −18.6541 −0.797593 −0.398796 0.917039i \(-0.630572\pi\)
−0.398796 + 0.917039i \(0.630572\pi\)
\(548\) 1.72740 0.0737907
\(549\) −11.1206 −0.474615
\(550\) 1.00000 0.0426401
\(551\) −7.45214 −0.317472
\(552\) −0.236307 −0.0100579
\(553\) 34.4381 1.46446
\(554\) 26.4077 1.12196
\(555\) 0.955534 0.0405602
\(556\) −0.527955 −0.0223903
\(557\) 15.6159 0.661668 0.330834 0.943689i \(-0.392670\pi\)
0.330834 + 0.943689i \(0.392670\pi\)
\(558\) 12.3954 0.524740
\(559\) 5.53942 0.234292
\(560\) −2.53237 −0.107012
\(561\) 0.143919 0.00607629
\(562\) 4.35199 0.183577
\(563\) −7.18692 −0.302892 −0.151446 0.988466i \(-0.548393\pi\)
−0.151446 + 0.988466i \(0.548393\pi\)
\(564\) 0.422998 0.0178114
\(565\) −18.8176 −0.791662
\(566\) 30.9015 1.29889
\(567\) 22.6444 0.950977
\(568\) −9.33354 −0.391627
\(569\) −5.13595 −0.215310 −0.107655 0.994188i \(-0.534334\pi\)
−0.107655 + 0.994188i \(0.534334\pi\)
\(570\) −0.108523 −0.00454553
\(571\) −18.9978 −0.795034 −0.397517 0.917595i \(-0.630128\pi\)
−0.397517 + 0.917595i \(0.630128\pi\)
\(572\) 5.53942 0.231615
\(573\) 1.39148 0.0581300
\(574\) −21.3322 −0.890390
\(575\) 2.94296 0.122730
\(576\) −2.99355 −0.124731
\(577\) −39.3920 −1.63991 −0.819954 0.572429i \(-0.806001\pi\)
−0.819954 + 0.572429i \(0.806001\pi\)
\(578\) −13.7874 −0.573481
\(579\) 1.48840 0.0618557
\(580\) −5.51381 −0.228948
\(581\) 5.26443 0.218405
\(582\) 0.910955 0.0377603
\(583\) 3.73022 0.154490
\(584\) 14.5726 0.603017
\(585\) 16.5825 0.685604
\(586\) 19.5789 0.808796
\(587\) −28.1015 −1.15987 −0.579936 0.814662i \(-0.696922\pi\)
−0.579936 + 0.814662i \(0.696922\pi\)
\(588\) 0.0471438 0.00194418
\(589\) 5.59634 0.230593
\(590\) 6.86311 0.282550
\(591\) −1.29871 −0.0534219
\(592\) 11.9002 0.489094
\(593\) 24.7093 1.01469 0.507345 0.861743i \(-0.330627\pi\)
0.507345 + 0.861743i \(0.330627\pi\)
\(594\) 0.481257 0.0197462
\(595\) 4.53893 0.186078
\(596\) 0.606138 0.0248284
\(597\) −1.55103 −0.0634794
\(598\) 16.3023 0.666651
\(599\) 32.9405 1.34591 0.672956 0.739683i \(-0.265024\pi\)
0.672956 + 0.739683i \(0.265024\pi\)
\(600\) −0.0802958 −0.00327806
\(601\) −31.7248 −1.29408 −0.647042 0.762455i \(-0.723994\pi\)
−0.647042 + 0.762455i \(0.723994\pi\)
\(602\) 2.53237 0.103212
\(603\) −33.0809 −1.34716
\(604\) −14.0485 −0.571625
\(605\) −1.00000 −0.0406558
\(606\) 0.874892 0.0355400
\(607\) 36.6188 1.48631 0.743156 0.669118i \(-0.233328\pi\)
0.743156 + 0.669118i \(0.233328\pi\)
\(608\) −1.35154 −0.0548122
\(609\) −1.12117 −0.0454320
\(610\) −3.71484 −0.150410
\(611\) −29.1816 −1.18056
\(612\) 5.36554 0.216889
\(613\) 44.1965 1.78508 0.892539 0.450969i \(-0.148922\pi\)
0.892539 + 0.450969i \(0.148922\pi\)
\(614\) 23.5744 0.951384
\(615\) −0.676398 −0.0272750
\(616\) 2.53237 0.102032
\(617\) −12.6211 −0.508105 −0.254053 0.967190i \(-0.581764\pi\)
−0.254053 + 0.967190i \(0.581764\pi\)
\(618\) −0.493831 −0.0198648
\(619\) 12.0355 0.483746 0.241873 0.970308i \(-0.422238\pi\)
0.241873 + 0.970308i \(0.422238\pi\)
\(620\) 4.14071 0.166295
\(621\) 1.41632 0.0568350
\(622\) −27.0824 −1.08591
\(623\) −11.1851 −0.448120
\(624\) −0.444792 −0.0178059
\(625\) 1.00000 0.0400000
\(626\) −15.2582 −0.609842
\(627\) 0.108523 0.00433399
\(628\) −12.7865 −0.510235
\(629\) −21.3295 −0.850462
\(630\) 7.58077 0.302025
\(631\) 10.4654 0.416619 0.208310 0.978063i \(-0.433204\pi\)
0.208310 + 0.978063i \(0.433204\pi\)
\(632\) 13.5992 0.540947
\(633\) −0.908886 −0.0361250
\(634\) 4.50710 0.179000
\(635\) 1.63100 0.0647243
\(636\) −0.299521 −0.0118768
\(637\) −3.25234 −0.128862
\(638\) 5.51381 0.218294
\(639\) 27.9404 1.10531
\(640\) −1.00000 −0.0395285
\(641\) 25.7943 1.01881 0.509406 0.860526i \(-0.329865\pi\)
0.509406 + 0.860526i \(0.329865\pi\)
\(642\) −1.26952 −0.0501040
\(643\) −20.7620 −0.818772 −0.409386 0.912361i \(-0.634257\pi\)
−0.409386 + 0.912361i \(0.634257\pi\)
\(644\) 7.45265 0.293676
\(645\) 0.0802958 0.00316164
\(646\) 2.42246 0.0953103
\(647\) −44.7180 −1.75805 −0.879024 0.476778i \(-0.841805\pi\)
−0.879024 + 0.476778i \(0.841805\pi\)
\(648\) 8.94201 0.351276
\(649\) −6.86311 −0.269400
\(650\) 5.53942 0.217274
\(651\) 0.841965 0.0329992
\(652\) 1.14615 0.0448866
\(653\) −12.5626 −0.491612 −0.245806 0.969319i \(-0.579053\pi\)
−0.245806 + 0.969319i \(0.579053\pi\)
\(654\) −0.572586 −0.0223899
\(655\) −15.3193 −0.598576
\(656\) −8.42383 −0.328895
\(657\) −43.6237 −1.70192
\(658\) −13.3405 −0.520066
\(659\) 18.5262 0.721679 0.360839 0.932628i \(-0.382490\pi\)
0.360839 + 0.932628i \(0.382490\pi\)
\(660\) 0.0802958 0.00312551
\(661\) 34.3543 1.33623 0.668114 0.744059i \(-0.267102\pi\)
0.668114 + 0.744059i \(0.267102\pi\)
\(662\) −15.3588 −0.596938
\(663\) 0.797230 0.0309619
\(664\) 2.07886 0.0806753
\(665\) 3.42259 0.132723
\(666\) −35.6238 −1.38039
\(667\) 16.2269 0.628309
\(668\) 18.8684 0.730041
\(669\) −0.653327 −0.0252591
\(670\) −11.0507 −0.426927
\(671\) 3.71484 0.143410
\(672\) −0.203338 −0.00784394
\(673\) −28.8322 −1.11140 −0.555699 0.831384i \(-0.687549\pi\)
−0.555699 + 0.831384i \(0.687549\pi\)
\(674\) −20.9110 −0.805463
\(675\) 0.481257 0.0185236
\(676\) 17.6852 0.680199
\(677\) 5.20811 0.200164 0.100082 0.994979i \(-0.468090\pi\)
0.100082 + 0.994979i \(0.468090\pi\)
\(678\) −1.51097 −0.0580286
\(679\) −28.7297 −1.10254
\(680\) 1.79237 0.0687341
\(681\) −2.12764 −0.0815315
\(682\) −4.14071 −0.158556
\(683\) −30.0087 −1.14825 −0.574125 0.818768i \(-0.694658\pi\)
−0.574125 + 0.818768i \(0.694658\pi\)
\(684\) 4.04591 0.154699
\(685\) −1.72740 −0.0660004
\(686\) −19.2134 −0.733570
\(687\) −1.66904 −0.0636779
\(688\) 1.00000 0.0381246
\(689\) 20.6632 0.787207
\(690\) 0.236307 0.00899607
\(691\) 20.1555 0.766750 0.383375 0.923593i \(-0.374762\pi\)
0.383375 + 0.923593i \(0.374762\pi\)
\(692\) −21.7578 −0.827107
\(693\) −7.58077 −0.287970
\(694\) −15.2696 −0.579626
\(695\) 0.527955 0.0200265
\(696\) −0.442736 −0.0167818
\(697\) 15.0986 0.571900
\(698\) −7.74515 −0.293158
\(699\) −1.98007 −0.0748931
\(700\) 2.53237 0.0957144
\(701\) −38.1038 −1.43916 −0.719580 0.694409i \(-0.755666\pi\)
−0.719580 + 0.694409i \(0.755666\pi\)
\(702\) 2.66588 0.100617
\(703\) −16.0836 −0.606603
\(704\) 1.00000 0.0376889
\(705\) −0.422998 −0.0159310
\(706\) −27.2898 −1.02707
\(707\) −27.5923 −1.03772
\(708\) 0.551078 0.0207108
\(709\) 4.17569 0.156821 0.0784107 0.996921i \(-0.475015\pi\)
0.0784107 + 0.996921i \(0.475015\pi\)
\(710\) 9.33354 0.350281
\(711\) −40.7099 −1.52674
\(712\) −4.41684 −0.165528
\(713\) −12.1859 −0.456367
\(714\) 0.364457 0.0136395
\(715\) −5.53942 −0.207163
\(716\) 10.6020 0.396214
\(717\) 2.04120 0.0762299
\(718\) −22.5389 −0.841143
\(719\) −39.6845 −1.47998 −0.739990 0.672617i \(-0.765170\pi\)
−0.739990 + 0.672617i \(0.765170\pi\)
\(720\) 2.99355 0.111563
\(721\) 15.5744 0.580022
\(722\) −17.1733 −0.639125
\(723\) −0.0272865 −0.00101480
\(724\) −21.1772 −0.787043
\(725\) 5.51381 0.204778
\(726\) −0.0802958 −0.00298006
\(727\) −30.1268 −1.11734 −0.558671 0.829390i \(-0.688688\pi\)
−0.558671 + 0.829390i \(0.688688\pi\)
\(728\) 14.0278 0.519906
\(729\) −26.6525 −0.987128
\(730\) −14.5726 −0.539355
\(731\) −1.79237 −0.0662931
\(732\) −0.298286 −0.0110250
\(733\) 40.0045 1.47760 0.738799 0.673925i \(-0.235393\pi\)
0.738799 + 0.673925i \(0.235393\pi\)
\(734\) −3.64335 −0.134479
\(735\) −0.0471438 −0.00173892
\(736\) 2.94296 0.108479
\(737\) 11.0507 0.407059
\(738\) 25.2172 0.928257
\(739\) −27.5323 −1.01279 −0.506396 0.862301i \(-0.669022\pi\)
−0.506396 + 0.862301i \(0.669022\pi\)
\(740\) −11.9002 −0.437459
\(741\) 0.601155 0.0220840
\(742\) 9.44627 0.346784
\(743\) −29.9996 −1.10058 −0.550290 0.834974i \(-0.685483\pi\)
−0.550290 + 0.834974i \(0.685483\pi\)
\(744\) 0.332482 0.0121894
\(745\) −0.606138 −0.0222072
\(746\) −22.3092 −0.816797
\(747\) −6.22317 −0.227694
\(748\) −1.79237 −0.0655354
\(749\) 40.0381 1.46296
\(750\) 0.0802958 0.00293199
\(751\) 9.62663 0.351281 0.175640 0.984454i \(-0.443800\pi\)
0.175640 + 0.984454i \(0.443800\pi\)
\(752\) −5.26799 −0.192104
\(753\) 1.12979 0.0411718
\(754\) 30.5433 1.11232
\(755\) 14.0485 0.511277
\(756\) 1.21872 0.0443244
\(757\) −29.3151 −1.06548 −0.532738 0.846280i \(-0.678837\pi\)
−0.532738 + 0.846280i \(0.678837\pi\)
\(758\) −4.46027 −0.162004
\(759\) −0.236307 −0.00857742
\(760\) 1.35154 0.0490255
\(761\) −16.6133 −0.602231 −0.301115 0.953588i \(-0.597359\pi\)
−0.301115 + 0.953588i \(0.597359\pi\)
\(762\) 0.130963 0.00474427
\(763\) 18.0582 0.653751
\(764\) −17.3295 −0.626958
\(765\) −5.36554 −0.193992
\(766\) −20.1980 −0.729782
\(767\) −38.0176 −1.37274
\(768\) −0.0802958 −0.00289742
\(769\) −18.2072 −0.656569 −0.328285 0.944579i \(-0.606470\pi\)
−0.328285 + 0.944579i \(0.606470\pi\)
\(770\) −2.53237 −0.0912601
\(771\) −0.260423 −0.00937890
\(772\) −18.5364 −0.667140
\(773\) 14.2373 0.512080 0.256040 0.966666i \(-0.417582\pi\)
0.256040 + 0.966666i \(0.417582\pi\)
\(774\) −2.99355 −0.107601
\(775\) −4.14071 −0.148739
\(776\) −11.3450 −0.407262
\(777\) −2.41976 −0.0868085
\(778\) 19.0027 0.681279
\(779\) 11.3851 0.407915
\(780\) 0.444792 0.0159261
\(781\) −9.33354 −0.333980
\(782\) −5.27487 −0.188629
\(783\) 2.65356 0.0948305
\(784\) −0.587126 −0.0209688
\(785\) 12.7865 0.456368
\(786\) −1.23008 −0.0438754
\(787\) −26.1616 −0.932562 −0.466281 0.884637i \(-0.654406\pi\)
−0.466281 + 0.884637i \(0.654406\pi\)
\(788\) 16.1741 0.576178
\(789\) 1.67608 0.0596702
\(790\) −13.5992 −0.483837
\(791\) 47.6530 1.69435
\(792\) −2.99355 −0.106371
\(793\) 20.5781 0.730749
\(794\) −1.53517 −0.0544811
\(795\) 0.299521 0.0106229
\(796\) 19.3164 0.684653
\(797\) −41.3388 −1.46430 −0.732148 0.681146i \(-0.761482\pi\)
−0.732148 + 0.681146i \(0.761482\pi\)
\(798\) 0.274820 0.00972852
\(799\) 9.44217 0.334040
\(800\) 1.00000 0.0353553
\(801\) 13.2221 0.467178
\(802\) 9.82030 0.346767
\(803\) 14.5726 0.514254
\(804\) −0.887327 −0.0312936
\(805\) −7.45265 −0.262672
\(806\) −22.9371 −0.807926
\(807\) −0.608043 −0.0214041
\(808\) −10.8959 −0.383315
\(809\) 31.9418 1.12301 0.561507 0.827472i \(-0.310222\pi\)
0.561507 + 0.827472i \(0.310222\pi\)
\(810\) −8.94201 −0.314190
\(811\) −30.1792 −1.05974 −0.529868 0.848080i \(-0.677759\pi\)
−0.529868 + 0.848080i \(0.677759\pi\)
\(812\) 13.9630 0.490004
\(813\) 0.964584 0.0338294
\(814\) 11.9002 0.417101
\(815\) −1.14615 −0.0401478
\(816\) 0.143919 0.00503819
\(817\) −1.35154 −0.0472844
\(818\) 20.0718 0.701793
\(819\) −41.9931 −1.46736
\(820\) 8.42383 0.294173
\(821\) 42.0871 1.46885 0.734425 0.678690i \(-0.237452\pi\)
0.734425 + 0.678690i \(0.237452\pi\)
\(822\) −0.138703 −0.00483781
\(823\) −15.3269 −0.534263 −0.267131 0.963660i \(-0.586076\pi\)
−0.267131 + 0.963660i \(0.586076\pi\)
\(824\) 6.15015 0.214250
\(825\) −0.0802958 −0.00279554
\(826\) −17.3799 −0.604724
\(827\) −43.1924 −1.50195 −0.750973 0.660333i \(-0.770415\pi\)
−0.750973 + 0.660333i \(0.770415\pi\)
\(828\) −8.80991 −0.306166
\(829\) −45.4509 −1.57857 −0.789287 0.614025i \(-0.789550\pi\)
−0.789287 + 0.614025i \(0.789550\pi\)
\(830\) −2.07886 −0.0721582
\(831\) −2.12043 −0.0735569
\(832\) 5.53942 0.192045
\(833\) 1.05235 0.0364616
\(834\) 0.0423926 0.00146794
\(835\) −18.8684 −0.652969
\(836\) −1.35154 −0.0467440
\(837\) −1.99275 −0.0688794
\(838\) −34.7946 −1.20196
\(839\) −37.8768 −1.30765 −0.653826 0.756645i \(-0.726837\pi\)
−0.653826 + 0.756645i \(0.726837\pi\)
\(840\) 0.203338 0.00701584
\(841\) 1.40208 0.0483475
\(842\) −10.3465 −0.356565
\(843\) −0.349446 −0.0120356
\(844\) 11.3192 0.389624
\(845\) −17.6852 −0.608388
\(846\) 15.7700 0.542184
\(847\) 2.53237 0.0870131
\(848\) 3.73022 0.128096
\(849\) −2.48126 −0.0851568
\(850\) −1.79237 −0.0614777
\(851\) 35.0218 1.20053
\(852\) 0.749444 0.0256755
\(853\) −22.8492 −0.782341 −0.391171 0.920318i \(-0.627930\pi\)
−0.391171 + 0.920318i \(0.627930\pi\)
\(854\) 9.40734 0.321912
\(855\) −4.04591 −0.138367
\(856\) 15.8106 0.540394
\(857\) −23.5090 −0.803052 −0.401526 0.915848i \(-0.631520\pi\)
−0.401526 + 0.915848i \(0.631520\pi\)
\(858\) −0.444792 −0.0151849
\(859\) 3.44951 0.117696 0.0588479 0.998267i \(-0.481257\pi\)
0.0588479 + 0.998267i \(0.481257\pi\)
\(860\) −1.00000 −0.0340997
\(861\) 1.71289 0.0583750
\(862\) −10.2756 −0.349990
\(863\) −42.4244 −1.44414 −0.722071 0.691819i \(-0.756810\pi\)
−0.722071 + 0.691819i \(0.756810\pi\)
\(864\) 0.481257 0.0163727
\(865\) 21.7578 0.739787
\(866\) 24.6166 0.836506
\(867\) 1.10707 0.0375981
\(868\) −10.4858 −0.355911
\(869\) 13.5992 0.461321
\(870\) 0.442736 0.0150101
\(871\) 61.2146 2.07418
\(872\) 7.13096 0.241485
\(873\) 33.9618 1.14943
\(874\) −3.97753 −0.134542
\(875\) −2.53237 −0.0856096
\(876\) −1.17012 −0.0395345
\(877\) 23.4047 0.790320 0.395160 0.918612i \(-0.370689\pi\)
0.395160 + 0.918612i \(0.370689\pi\)
\(878\) 2.26151 0.0763223
\(879\) −1.57210 −0.0530256
\(880\) −1.00000 −0.0337100
\(881\) −31.0051 −1.04459 −0.522295 0.852765i \(-0.674924\pi\)
−0.522295 + 0.852765i \(0.674924\pi\)
\(882\) 1.75759 0.0591812
\(883\) −19.9759 −0.672243 −0.336122 0.941819i \(-0.609115\pi\)
−0.336122 + 0.941819i \(0.609115\pi\)
\(884\) −9.92867 −0.333937
\(885\) −0.551078 −0.0185243
\(886\) −28.6143 −0.961315
\(887\) 42.7824 1.43649 0.718246 0.695790i \(-0.244945\pi\)
0.718246 + 0.695790i \(0.244945\pi\)
\(888\) −0.955534 −0.0320656
\(889\) −4.13029 −0.138526
\(890\) 4.41684 0.148053
\(891\) 8.94201 0.299569
\(892\) 8.13650 0.272430
\(893\) 7.11990 0.238259
\(894\) −0.0486703 −0.00162778
\(895\) −10.6020 −0.354385
\(896\) 2.53237 0.0846004
\(897\) −1.30901 −0.0437064
\(898\) 10.5148 0.350882
\(899\) −22.8311 −0.761459
\(900\) −2.99355 −0.0997851
\(901\) −6.68592 −0.222740
\(902\) −8.42383 −0.280483
\(903\) −0.203338 −0.00676667
\(904\) 18.8176 0.625864
\(905\) 21.1772 0.703953
\(906\) 1.12803 0.0374764
\(907\) −6.88502 −0.228613 −0.114307 0.993446i \(-0.536465\pi\)
−0.114307 + 0.993446i \(0.536465\pi\)
\(908\) 26.4976 0.879353
\(909\) 32.6173 1.08185
\(910\) −14.0278 −0.465018
\(911\) −14.2895 −0.473433 −0.236716 0.971579i \(-0.576071\pi\)
−0.236716 + 0.971579i \(0.576071\pi\)
\(912\) 0.108523 0.00359356
\(913\) 2.07886 0.0688001
\(914\) 34.8469 1.15263
\(915\) 0.298286 0.00986104
\(916\) 20.7862 0.686794
\(917\) 38.7942 1.28110
\(918\) −0.862589 −0.0284697
\(919\) 21.1108 0.696380 0.348190 0.937424i \(-0.386796\pi\)
0.348190 + 0.937424i \(0.386796\pi\)
\(920\) −2.94296 −0.0970266
\(921\) −1.89292 −0.0623739
\(922\) −20.8632 −0.687092
\(923\) −51.7024 −1.70180
\(924\) −0.203338 −0.00668934
\(925\) 11.9002 0.391275
\(926\) 7.25956 0.238564
\(927\) −18.4108 −0.604690
\(928\) 5.51381 0.181000
\(929\) 15.0036 0.492252 0.246126 0.969238i \(-0.420842\pi\)
0.246126 + 0.969238i \(0.420842\pi\)
\(930\) −0.332482 −0.0109025
\(931\) 0.793525 0.0260067
\(932\) 24.6597 0.807754
\(933\) 2.17460 0.0711934
\(934\) −14.0704 −0.460397
\(935\) 1.79237 0.0586167
\(936\) −16.5825 −0.542017
\(937\) −2.66952 −0.0872096 −0.0436048 0.999049i \(-0.513884\pi\)
−0.0436048 + 0.999049i \(0.513884\pi\)
\(938\) 27.9845 0.913726
\(939\) 1.22517 0.0399820
\(940\) 5.26799 0.171823
\(941\) 10.9184 0.355931 0.177965 0.984037i \(-0.443048\pi\)
0.177965 + 0.984037i \(0.443048\pi\)
\(942\) 1.02670 0.0334516
\(943\) −24.7910 −0.807306
\(944\) −6.86311 −0.223375
\(945\) −1.21872 −0.0396449
\(946\) 1.00000 0.0325128
\(947\) 60.2218 1.95695 0.978474 0.206372i \(-0.0661658\pi\)
0.978474 + 0.206372i \(0.0661658\pi\)
\(948\) −1.09196 −0.0354651
\(949\) 80.7235 2.62040
\(950\) −1.35154 −0.0438498
\(951\) −0.361901 −0.0117354
\(952\) −4.53893 −0.147107
\(953\) −20.6439 −0.668721 −0.334361 0.942445i \(-0.608520\pi\)
−0.334361 + 0.942445i \(0.608520\pi\)
\(954\) −11.1666 −0.361532
\(955\) 17.3295 0.560768
\(956\) −25.4210 −0.822173
\(957\) −0.442736 −0.0143116
\(958\) 37.2717 1.20420
\(959\) 4.37440 0.141257
\(960\) 0.0802958 0.00259154
\(961\) −13.8545 −0.446920
\(962\) 65.9201 2.12535
\(963\) −47.3298 −1.52518
\(964\) 0.339825 0.0109450
\(965\) 18.5364 0.596709
\(966\) −0.598417 −0.0192537
\(967\) 49.1054 1.57912 0.789561 0.613672i \(-0.210308\pi\)
0.789561 + 0.613672i \(0.210308\pi\)
\(968\) 1.00000 0.0321412
\(969\) −0.194513 −0.00624866
\(970\) 11.3450 0.364266
\(971\) 3.95375 0.126882 0.0634410 0.997986i \(-0.479793\pi\)
0.0634410 + 0.997986i \(0.479793\pi\)
\(972\) −2.16178 −0.0693390
\(973\) −1.33698 −0.0428615
\(974\) −27.3833 −0.877418
\(975\) −0.444792 −0.0142447
\(976\) 3.71484 0.118909
\(977\) 38.7865 1.24089 0.620445 0.784250i \(-0.286952\pi\)
0.620445 + 0.784250i \(0.286952\pi\)
\(978\) −0.0920309 −0.00294282
\(979\) −4.41684 −0.141163
\(980\) 0.587126 0.0187551
\(981\) −21.3469 −0.681554
\(982\) 18.8903 0.602814
\(983\) 43.7599 1.39573 0.697863 0.716232i \(-0.254135\pi\)
0.697863 + 0.716232i \(0.254135\pi\)
\(984\) 0.676398 0.0215628
\(985\) −16.1741 −0.515350
\(986\) −9.88277 −0.314731
\(987\) 1.07118 0.0340962
\(988\) −7.48675 −0.238185
\(989\) 2.94296 0.0935807
\(990\) 2.99355 0.0951413
\(991\) 21.9423 0.697019 0.348509 0.937305i \(-0.386688\pi\)
0.348509 + 0.937305i \(0.386688\pi\)
\(992\) −4.14071 −0.131468
\(993\) 1.23325 0.0391360
\(994\) −23.6359 −0.749686
\(995\) −19.3164 −0.612372
\(996\) −0.166923 −0.00528917
\(997\) −51.3411 −1.62599 −0.812994 0.582272i \(-0.802164\pi\)
−0.812994 + 0.582272i \(0.802164\pi\)
\(998\) 7.35085 0.232687
\(999\) 5.72704 0.181196
Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))

Twists

       By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 4730.2.a.bf.1.7 13
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
4730.2.a.bf.1.7 13 1.1 even 1 trivial